JP4010362B2 - Quantum Turing machine - Google Patents

Description

  The present invention relates to a quantum Turing machine formed using a superconductor.

Prior to the present invention, superconducting electronics characterized by using phase difference solitons generated between two or more types of superconducting electrons as a technique for applying the phase difference between a plurality of superconducting order parameters to electronics. Patent Document 1 below (hereinafter referred to as “previous invention”).
JP 2003-209301 A

  On the other hand, various quantum bits have been devised in the quantum Turing machine. As a typical example, there are a method using a nuclear spin and a method using an atomic energy level. Attempts have also been made to create artificial atoms from semiconductors. Qubits using ordinary superconductors have also been proposed.

  In the previous invention, the inter-band phase difference soliton S, which is an information bit, is ubiquitous on the connected circuit, and thus its operation is complicated. It is necessary to confine this in a spatially narrow area to facilitate bit manipulation.

  In the previous invention, a method for forming a quantum superposition state and forming a qubit was provided, but a basic calculation method for configuring a quantum Turing machine using this qubit was not provided. .

  To date, in the qubits proposed to make quantum Turing machines, any of the proposed methods has been difficult to increase in number of bits with the current technology, and more than 100 years have been required for practical use. It is supposed to take.

  In addition, in qubits that do not use macroscopic quantum states such as superconductivity, the superposition state for functioning the quantum Turing machine tends to collapse due to interaction with the outside world, and the quantum algorithm can be changed using the quantum Turing machine. I don't get enough time to do it.

  The present invention has been proposed in view of the above, and it is possible to easily configure qubits, to surely execute basic logical operations, to increase the number of bits, and to further provide a quantum algorithm. It is an object of the present invention to provide a quantum Turing machine capable of securing a sufficient time for executing the above.

In order to achieve the above object, the invention according to claim 1 is a quantum Turing machine formed using a superconductor, wherein a plurality of types of superconducting electrons present in each of a plurality of bands of the superconductor. It consists of qubits made using phase difference solitons generated between them, and the qubits are narrowed at least in two places: the linear circuit body formed of the superconductor and the linear circuit body. linear circuit and a well-shaped portion formed Te, to configure by localizing the phase difference soliton is characterized by.

According to a second aspect of the present invention, there is provided a phase difference soliton generated between a plurality of types of superconducting electrons in each of a plurality of bands of the superconductor in a quantum Turing machine formed using the superconductor. The qubit includes a ring body formed of the superconductor and a well-shaped portion formed by reducing the diameter at least at two locations of the ring body. the ring, to configure by localizing the phase difference soliton is characterized by.

The present invention as described in claim 3, in addition to the configuration of the invention according to claim 2 described above, one of the two wells shaped portion, the first having a first well-shaped portion in a part thereof An auxiliary ring is added to the ring, a second auxiliary ring having a second well-like portion in part is added to the ring, and a switch is provided for each of the ring, the first auxiliary ring, and the second auxiliary ring. It is characterized in that a quantum bit is provided.

According to a fourth aspect of the invention, in addition to the configuration of the second aspect of the invention described above, the phase difference soliton is localized in the ring, and each of the first auxiliary ring and the second auxiliary ring is provided. It is characterized by realizing unitary conversion of qubits by operating a switch.

Further, an invention according to claim 5, in addition to the structure of the invention according to claim 4 as described above, unitary transformation of the qubit, first in the ring, which corresponds to a phase slip by the phase difference soliton field Turn on the ring switch to make a phase difference soliton in the ring, and then turn on the selection auxiliary ring of either the first or second auxiliary ring, and in the selection auxiliary ring It is characterized in that the magnetic field is applied as an external field, and then the selection auxiliary ring is turned off and then turned on again after a predetermined time has elapsed.

According to a sixth aspect of the present invention, in addition to the configuration of the third aspect of the present invention, two qubits are arranged in parallel, and a switch is provided in the first auxiliary ring of one qubit. And an interaction ring provided with a switch is overlapped over both the second auxiliary ring of one qubit and the second auxiliary ring of the other qubit. A control NOT gate between bits is formed.

The invention described in Claim 7, in addition to the structure of the invention according to claim 6 as described above, calculation of the controlled-NOT between the two qubits, first in each of the two rings of a qubit, It is characterized in that it is executed by turning on a switch of each ring while applying a magnetic field corresponding to a phase slip caused by a phase difference soliton to create a phase difference soliton, and then turning on an interaction ring.

  In the quantum Turing machine of the present invention, by devising the shape of the superconducting electric wire, the soliton S can be localized in the superconductor having a plurality of superconducting order parameters, and the bit operation is simplified. A qubit can be easily configured. In addition, basic logical operation processing can be surely executed. Further, multi-bit and integration are possible. Furthermore, by using a macroscopic quantum state called superconductivity, coherence can be easily maintained, and sufficient time can be secured for executing the quantum algorithm.

Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings.
In the present invention, a quantum Turing machine is constituted by qubits made by using the phase difference between superconducting order parameters existing in each of a plurality of bands of a superconductor. The use of the phase difference between the superconducting order parameters is, for example, use of a phase difference soliton S generated between a plurality of types of superconducting electrons.

First, a configuration for localizing the phase difference soliton S will be described. As shown in FIG. 1A, a ring R made of a superconductor having a plurality of bands is opened and closed by a switch SW. Examples of the superconductor having a plurality of bands include Cu _x Ba ₂ Ca ₃ CuO _y .

  When the switch SW is turned on while applying the magnetic field Φsoliton corresponding to the phase slip Θsoliton due to the interband phase difference soliton S, the soliton S is generated in the ring R (FIG. 1B). Here, the soliton S is indicated by an arrow. The direction of phase shift is indicated by the direction of the arrow.

  The soliton S produced in this way is ubiquitous in the ring R because the energy is the same anywhere in the ring R. Its existence probability follows a quantum mechanical probability distribution.

  Next, an example of a shape for localizing the soliton S will be described. In FIG. 1, the circuit (ring) is represented by an electric wire having no thickness, but the superconducting electric wire actually constituting the circuit has a thickness. The conceptual diagram of the electric wire is shown in FIG. If one soliton S is placed on the electric wire C, the soliton S is ubiquitous in FIG. Since the energy of the soliton S is proportional to the cross-sectional area of the electric wire C, if a thin portion is formed in the electric wire C as shown in FIG. 2B, the soliton S is likely to be fitted (localized easily). FIG. 2 (c) conceptually shows this state with the potential energy felt by the soliton S as the vertical axis and the direction in which the wire C extends as the horizontal axis.

  Thus, the soliton S can be localized by processing such as partially thinning the electric wire C.

  Next, an example in which a qubit is configured using the soliton S localized by the above method will be described.

As shown to Fig.3 (a), the part (well-like part) W1, W2 made thin to the electric wire C is made in two places on right and left. The potential produced by the electric wire C is illustrated in FIG. If one soliton S is put here, it will classically be either left (left well portion W1) or right (right well portion W2). This is illustrated in FIGS. 4 (a) and 4 (b). Here, the soliton S is represented by a black circle. Quantum mechanically, this problem corresponds to a state where one particle is put in a system having two well-type potentials. Soliton S corresponds to particles. Soliton S can tunnel through the potential barrier. The generation operator that creates the soliton S on the left is a ⁺ _L , the annihilation operator is a _L , the generation operator that creates the soliton S is a ⁺ _R , the annihilation operator is a _R , and the tunnel probability is T _A , Assuming that the depth of the well-type potential is V _L and V _R as shown in FIG. 4A, the Hamiltonian H of this system is expressed by the following equation (1).

When one soliton S is introduced into such a Hamiltonian H, it becomes a qubit. When the wave function when the soliton S is on the left is ψ _L , and the wave function when the soliton S is on the right is ψ _R , the wave function ψ (t) of this qubit is expressed by the following equation (2). .

In the above equation (2), c _L (t) and c _R (t) are coefficients that change with time, are complex numbers, and satisfy the following equation (3).

By the way, when realizing a quantum Turing machine using a qubit, the properties required for the qubit are:
(1) Arbitrary unitary transformation can be applied to one qubit. (2) The control NOT gate can be configured with two qubits (by Tetsuro Nishino, Iwanami Lecture, Physics World, “Quantum Computer and Quantum Cipher "issued on March 15, 2002, ISBN4-00-011159-0, page 36). A quantum Turing machine that satisfies these two points was realized with a qubit configured using the above-described localized soliton S. An example is shown below.

  First, an example of a method for realizing unitary transformation of 1 qubit is shown.

As shown in FIGS. 5 (a) and 5 (b), the ring body R1 formed of a superconductor, the switch SW provided on the ring body R1, and the ring body R1 are formed with a reduced diameter. A ring R0 is formed by the portions (well-like portions) W1 and W2. Then, the soliton S is formed on the ring R0 by the same method as in FIG. In FIG. 5B, the soliton S is generated in either of the well-like portions W1 and W2. Also, leave the outside for a long time. At this stage, Hamiltonian H has the same form as equation (1) above. Further, the shape of the well-like portions W1, W2 is made the same. That is, V _L = V _R. At this time, the eigenfunction Ψ (t) of the Hamiltonian H is expressed by the following expression (4).

In the equation (4), c _A (t = 0) = 0 usually immediately after the switch is turned on (t = 0).

  Now, as shown in FIG. 6, among the two well-like portions W1 and W2 of the ring R0, the first auxiliary ring RL having the first well-like portion W1 as a part thereof is added to the ring R0. In addition, a second auxiliary ring RR having a second well-shaped portion W2 as a part thereof is added to the ring R0, and switches SW1 and SW2 are provided in each of the first auxiliary ring RL and the second auxiliary ring RR to provide a quantum bit. Configure.

だけいれることにする。 Then, as shown in FIG. 7, the switch SW1 of the first auxiliary ring RL is turned on, and a magnetic flux is generated as an external field in the first auxiliary ring RL.

I will just put it.

第２井戸状部分Ｗ２にソリトンＳがいる場合は、次式（７）が成立する。

From the boundary condition, when the soliton S is present in the first well-shaped portion W1 of the ring R0, the following equation (6) is established:

When the soliton S is present in the second well portion W2, the following equation (7) is established.

In the above formulas (6) and (7), μ _L is the self-induction coefficient of the first auxiliary ring RL, I _L is the induced superconducting current induced in the first auxiliary ring RL, and Φ soliton compensates for the phase slip caused by the soliton S. It is the amount of magnetic flux to do.

だけエネルギーが高くなり、ソリトンＳがない場合、

だけエネルギーが高くなる。 If there is a soliton S,

If the energy is only high and there is no soliton S,

Only the energy gets higher.

とおくと、系のハミルトニアンＨは、式（１）を使って、

となる。 here,

After that, the Hamiltonian H of the system is

It becomes.

となるようにすると、

とおいてあげて、系の波動関数Ψ（ｔ）は、

という形になる。

So that

For example, the wave function Ψ (t) of the system is

It becomes the form.

In the above equation (14), t _a is the time when the switch SW1 of the first auxiliary ring RL is turned on.

  Next, an example of how the temporal development of the wave function Ψ (t) occurs when the switch SW1 of the first auxiliary ring RL is turned on and off will be described.

In FIG. 7, when the switch SW of the ring R0 is turned on, t = 0. 0 In <t <t _a, rather than temporal evolution, precisely, in the phase difference in the coefficient of [psi _L and [psi _R does not occur, the following equation (15) holds.

となる。

となる時間ｔ＝ｔ_bまで第１補助リングＲＬのスイッチＳＷ１をいれておく。この時間になると

になるので、スイッチＳＷ１をオフにする。 Then at time t = t _a, it is assumed that the switch SW1 of the first auxiliary ring RL on. The temporal development at this time is

It becomes.

It should put the switch SW1 of the first auxiliary ring RL until become time t = t _b. At this time

Therefore, the switch SW1 is turned off.

The wave function Ψ (t) at t = t _b is expressed by the following equation (19).

とおくと、

となり、再び、ｔ＝ｔ_cに第１補助リングＲＬのスイッチＳＷ１をいれてあげると、それ以降の時間的発展は、

とおいて、

とおいてあげて、

となる。 The time evolution at t> t _b after turning off SW1 returns to the form of equation (4) again,

After all,

Then, when the switch SW1 of the first auxiliary ring RL is turned on again at t = t _c , the temporal development thereafter is

Anyway,

Let me give you

It becomes.

Since α ₀ and β can be arbitrarily selected between 0 and 2π, the above equation (24) provides n arbitrary unitary transformations for one qubit which is a constituent element of the quantum Turing machine. That is, FIG. 6 and FIG. 7 provide an example of one qubit which is a constituent element for configuring a quantum Turing machine. It is obvious that the second auxiliary ring RR on the right side has a similar function.

  Next, a method of creating a control NOT gate between two qubits necessary for making a quantum Turing machine will be described. The qubit described in FIGS. 6 and 7 is defined as a qubit B, and a qubit A having the same configuration as the qubit B is newly added thereto. This is shown in FIG. For simplicity, only the wires associated with the control NOT gate are left. Furthermore, an interaction ring M and a balance ring N for causing the qubits A and B to interact with each other were prepared.

  That is, as shown in FIG. 8, the two qubits A and B are arranged in parallel, and the balance ring N provided with the switch SW3 is superimposed on the first auxiliary ring RL of the qubit B, and further the qubit B A control NOT gate between two qubits is formed by overlapping the interaction ring M including the switch SW4 over both the second auxiliary ring RR and the second auxiliary ring RR of the qubit A.

  For simplicity, it is assumed here that the self-induction coefficient of the interaction ring M and the self-induction coefficient of the balance ring N are designed to be the same. Let the self-induction coefficient be μcoupling. In addition, the first auxiliary ring RL and the balance ring N of the qubit B are electrically insulated from each other, and are superposed on each other so that only the magnetic flux generated in the first auxiliary ring RL of the qubit B can be felt. Keep it.

  Both the interaction ring M and the balance ring N are composed of a superconductor having a single order parameter, and are superconducting so as to cancel all the magnetic fluxes generated on the auxiliary rings RL and RR of the qubits A and B. Allow current to flow. In other words, the magnetic flux passes through the auxiliary rings RL of the qubits A and B, is repelled from the interaction ring M, and passes along a path that goes out of the interaction ring M.

  The calculation of the control NOT between the two qubits A and B is performed by first applying each magnetic field corresponding to the phase slip Θsoliton caused by the phase difference soliton S to each ring R0 of the two qubits A and B. This is performed by creating a soliton S in R0 and then turning on the switch SW4 of the interaction ring M.

  The state realized at this time is a superposition state of the states listed in FIGS. In FIGS. 9 to 12, the switch SW3 of the balance ring N and the switch SW4 of the interaction ring M are omitted, and are omitted (the interaction starts when the switch is turned on).

Each state defined in FIGS. 9 to 12 is written as | 00>, | 01>, | 10>, | 11>, and the Hamiltonian when each of the states is from the first component to the fourth component. Is written down, the following equation (25) is obtained.

  Here, for simplicity, the outside field applied to the auxiliary rings RL and RR was cut off. Further, the auxiliary rings RL and RR of the qubits A and B are considered to have the same self-induction coefficient, and the energy generated by the annular superconducting current is deducted beforehand as a clog. The energy of the magnetic field due to the mutual induction coefficient was ignored for simplicity.

は、相互作用リングＭ、バランスリングＮで生じるエネルギーである。上記の行列式（２５）から、さらに下駄になっている、εを引き去ると、次式（２７）となる。

T _A is the tunnel probability of the soliton S at the qubit A, and T _B is the tunnel probability of the soliton S at the qubit B. Also,

Is energy generated in the interaction ring M and the balance ring N. When ε, which is a further clog, is subtracted from the determinant (25), the following expression (27) is obtained.

Next, assuming T _A << T _B , the following equation (28) is established.

T _A << T _B can be changed by changing the length of the electric wires constituting the qubits A and B before the qubits A and B interact with each other. Also, as will be described later, the tunnel probability can be changed using an auxiliary ring.

となる。 Furthermore, when T _B << ε,

It becomes.

The time evolution of the wave function Ψ (t) of the system is expressed by the following equation (30).

は複素数の定数である。特に、

の時、｜１０＞は｜１１＞に、｜１１＞は｜１０＞に変換される。 In the above equation (30),

Is a complex constant. In particular,

In this case, | 10> is converted to | 11>, and | 11> is converted to | 10>.

となる。ただし、この式には下記に示す位相項が付随している。

That is, at this time, the unitary transformation matrix for the wave function is

It becomes. However, this equation is accompanied by the following phase term.

を実現することができる。 This phase term can be set to 1 by the unitary transformation for one qubit described above. By applying this unitary conversion, the target control NOT gate is finally obtained.

Can be realized.

In the above, the setting that the tunnel probability is T _A << T _B is changed by changing the length of the qubit wires before making the qubits A and B interact with each other. However, it is possible to change the tunnel probability using an auxiliary ring. For example, as shown in FIG. 13, when the auxiliary ring R10 is attached and an external field is put therein, the potential barrier can be effectively changed, and the tunnel probability of the soliton S can be changed.

  In the above example, the number of qubits is one or two, but an arbitrary number of qubits can be provided to increase the number of bits.

  The bit can be read out by measuring the magnetic flux generated in the auxiliary rings RL and RR. Once read, the soliton S can be treated as a classical information bit by changing the site energy of the soliton S to be larger than the tunnel energy. That is, the probability that the phase difference soliton S is localized may be either 0% or 100%, and classical digital logic that does not use the superposition state may be performed.

  In addition, a plurality of superconducting order parameters used in the present invention may exist not only in a plurality of bands but also in the same band. For example, in a d-wave superconductor, as shown in FIG. 14, strong interaction works in the region shown up to the vertical stripe of the Fermi surface and in the region shown up to the horizontal stripe, and weak between the vertical stripe and the horizontal stripe. In superconductors in which interaction works, separate superconducting order parameters can be defined in the vertical stripes and horizontal stripes, and solitons S can exist in superconductors having the same band. This soliton S can also exhibit the configuration and effects disclosed in the present invention.

  As described above, in the quantum Turing machine of the present invention, the soliton S can be localized in a superconductor having a plurality of superconducting order parameters by devising the shape of the superconducting electric wire.

  Therefore, the bit operation is simplified, and the configuration of the qubit can be easily performed. Further, basic logical operation processing such as unitary conversion and control NOT operation can be reliably executed. Further, multi-bit and integration are possible. Furthermore, by using a macroscopic quantum state called superconductivity, coherence can be easily maintained, and sufficient time can be secured for executing the quantum algorithm.

It is explanatory drawing of the method of generating a soliton in a superconductor. It is a figure which shows the example of the shape of the electric wire for localizing a soliton, and the principle to localize. It is a figure which shows the method of comprising a qubit with two well-type potentials and solitons. It is a figure explaining the method of throwing one soliton in the well type potential of FIG. It is a figure explaining the condition which generates a soliton in a superconducting electric wire with two well type potentials. It is a figure for demonstrating one form which enables arbitrary unitary transformations with a qubit. It is a figure for demonstrating one form which enables arbitrary unitary transformations with a qubit. It is a figure for demonstrating one form which enables a control NOT gate using 2 qubits. It is a figure for demonstrating a mode that a control NOT gate functions in 2 qubits. It is a figure for demonstrating a mode that a control NOT gate functions in 2 qubits. It is a figure for demonstrating a mode that a control NOT gate functions in 2 qubits. It is a figure for demonstrating a mode that a control NOT gate functions in 2 qubits. It is a figure explaining one form which modulates the tunnel probability of the soliton which comprises a qubit with an electric method. It is a figure for demonstrating one form with a plurality of order parameters even in a single band.

Explanation of symbols

A qubit B qubit C wire M interaction ring N balance ring R0 ring R1 ring body R10 auxiliary ring RL first auxiliary ring RR second auxiliary ring S interband phase difference soliton SW ring R switch SW1 first auxiliary ring RL SW2 Switch of second auxiliary ring RR SW3 Switch of balance ring N SW4 Switch of interaction ring M W1 Well-shaped part W2 Well-shaped part

Claims (7)

In quantum Turing machines formed using superconductors,
Consists of qubits made using phase difference solitons generated between multiple types of superconducting electrons that exist in each of the multiple bands of the superconductor,
The qubit is
A linear circuit body formed of the superconductor;
A well-shaped portion formed by reducing the diameter in at least two locations of the linear circuit body;
It is configured by localizing a phase difference soliton in a linear circuit comprising
Quantum Turing machine characterized by that. In quantum Turing machines formed using superconductors,
Consists of qubits made using phase difference solitons generated between multiple types of superconducting electrons that exist in each of the multiple bands of the superconductor,
The qubit is
A ring body formed of the superconductor;
A well-shaped part formed by reducing the diameter in at least two locations of the ring body;
It is configured by localizing a phase difference soliton on a ring comprising
Quantum Turing machine characterized by that.   Of the two well-like parts, a first auxiliary ring having a first well-like part as a part thereof is added to the ring, and a second auxiliary ring having a second well-like part as a part thereof is provided. The quantum Turing machine according to claim 2, wherein a quantum bit is configured by adding a switch to each of the ring, the first auxiliary ring, and the second auxiliary ring to form a qubit.   4. The quantum Turing machine according to claim 3, wherein a phase difference soliton is localized in the ring, and unitary transformation of qubits is realized by operating each switch of the first auxiliary ring and the second auxiliary ring. 5.   In the unitary transformation of the qubit, first, while applying a magnetic field corresponding to a phase slip caused by a phase difference soliton in the ring, the ring is turned on to create a phase difference soliton in the ring, and then the first, second Turn on the selection auxiliary ring of one of the auxiliary rings, put magnetic flux as an external field in the selection auxiliary ring, then turn off the selection auxiliary ring, and turn it on again after a predetermined time The quantum Turing machine according to claim 4, wherein   Two qubits are arranged in parallel, and a balance ring with a switch is superimposed on the first auxiliary ring of one qubit, and the second auxiliary ring of one qubit and the other qubit 4. The quantum Turing machine according to claim 3, wherein a control NOT gate between two qubits is constructed by superimposing interaction rings with switches over both the second auxiliary ring.   The calculation of the control NOT between the two qubits is as follows. First, while applying a magnetic field corresponding to the phase slip caused by the phase difference soliton in each ring of the two qubits, the switch of each ring is turned on to set the phase difference soliton. 7. The quantum Turing machine according to claim 6, wherein the quantum Turing machine is executed by making and then switching on the interaction ring.

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